One of the difficulties with self consistent plate-mantle models capturing multiple physical features, such as elasticity, non-Newtonian flow properties, and temperature dependence, is that the individual behaviours cannot be considered in isolation. For instance, if a viscous mantle convection model is generalized naively to include hypo-elasticity, then problems based on Earth-like Rayleigh numbers exhibit almost insurmountable numerical stability issues due to spurious softening associated with the co-rotational stress terms. If a stress limiter is introduced in the form of a power law rheology or yield criterion these difficulties can be avoided. In this paper, a novel Eulerian finite element formulation for visco-elastic convection is presented and the implementation of the co-rotational stress terms is addressed. The salient dimensionless numbers of visco-elastic plastic flows such as Weissenberg, Deborah and Bingham numbers are discussed in a separate section in the context of Geodynamics. We present an Eulerian formulation for slow temperature dependent, visco-elastic-plastic flows. A consistent tangent (incremental) formulation of the governing equations is derived. Numerical and analytical solutions demonstrating the effect of visco-elasticity, co-rotational terms are first discussed for simplified benchmark problems. For flow around cylinders we identify parameter ranges of predominantly viscous and visco-plastic and transient behavior. The influence of locally high strain rates on the importance of elasticity and non-Newtonian effects is also discussed in this context. For the case of simple shear we investigate in detail the effect of different co-rotational stress rates and the effect of power law creep. The results show that the effect of the co-rotational terms is insignificant if realistic stress levels are considered (e.g. deviatoric invariant smaller than 1/10 of the shear modulus say). We also consider the basic convection modes of stagnant lid, episodic resurfacing and mobile lid convection as applicable to a cooling planet. The simulations show that elasticity does not have a significant effect on global parameters such as the Nusselt number and the qualitative nature of the basic convection pattern. Our simple benchmarks show, however, also that elasticity plays a significant role for instabilities on the local scale of an individual subduction zone.

Originally published as Muhlhaus, Hans-Bernd and Regenauer-Lieb, Klaus (2005) Towards a Self Consistent Plate Mantle Model that Includes Elasticity: Simple Benchmarks and Application to Basic Modes of Convection. Geophysical Journal International 163:788-800. doi:10.1111/j.1365-246X.2005.02742.x